In the previous chapters we described a systematic procedure to study the classical dynamics of a binary coalescing system using a Post-Newtonian expansion. On this chapter instead we want to develop an alternative, but complementary, framework, that allow us to obtain both classical and quantum General Relativistic corrections to the Newtonian potential between two massive objects. This is possible by working in the Post-Minkowskian perturbative scheme, which is a weak field expansion where at \(n\) -PM order we consider corrections that scale as \(G_N^n\) . By working in this relativistic framework, one obtains expressions valid at all order in velocity powers, and can recover post-Newtonian results by focusing on the classical contributions, and performing a suitable non-relativistic limit, as first noticed in 1971 by Iwasaki in [1]}. In order to develop this framework, we have to look again to the full theory of General Relativity, this time from a QFT perspective, considering it not as a fundamental theory but rather as a low-energy effective field theory (EFT)[2]}, [3]} of an unknown ultraviolet theory (UV) of gravity (see [4]}, [3]} for general reviews). Within the EFT approach, it is possible to study a binary coalescing system as a scattering process, involving particles that can have different spins, which are gravitationally interacting. Moreover, from these scattering amplitudes it is possible to compute post-Minkowskian corrections to the Newtonian potential between two massive objects.[6]}[7]}[8]}. In order to compute them, it is not necessary to evaluate the full loop amplitudes, but we can focus only on the long-range contributions which arise from non-analytical terms in the amplitude. In the treatment of the PM approach to GR we will adopt the "mostly minus" convention: \(\eta _{\mu \nu }=diag(+,-,-,-)\) .
